Note: the post on using cumulative momentum as a price proxy will be reserved for later due to its complexity.

The geometric mean is the average derived by compounding different momentums or price returns. In contrast the arithmetic mean is simple the average of different momentums or price returns. The geometric return is often quite different from the arithmetic mean return–this divergence is a source of valuable information. At the limit, the geometric mean and the arithmetic should converge. This means that if one strays too far from the other, it is likely to return to normal. This is especially true in shorter-term series, where positive divergence between the geometric mean and arithmetic mean is negative for future stock prices.In contrast the opposite effect occurs with longer term price series–where a positive divergence is actually favorable for future stock prices.. This excerpt is derived from wikipedia http://en.wikipedia.org/wiki/Arithmetic-geometric_mean:

From inequality of arithmetic and geometric means we can conclude that:

and thus

that is, the sequence gi is nondecreasing. Furthermore, it is easy to see that it is also bounded above by the larger of x and y (which follows from the fact that both arithmetic and geometric means of two numbers both lie between them). Thus, from Bolzano-Weierstrass theorem, there exists a convergent subsequence of gi. However, since the sequence is nondecreasing, we can conclude that the sequence itself is convergent, so there exists a g such that:

However, we can also see that:

and so:

Surprisingly, this simple concept underlies a short-term oscillator that actually outperforms  buying into weakness and selling into strength for the S&P500 all the way back to 1955 not including the 1997-2009 period!   The average weekly return prior to buy signals was -.3%, and sell signals were given when weekly returns were .42%.  When the oscillator was below .5 (oversold), the market returned 6% compounded from 1955-2009 not including dividends. When the oscillator was above .5 (overbought), the market made 0.0016%!

The creation of the oscillator involves taking the arithmetic average of the percentage returns (or 1-day ROCs) over the last 5 days.  A geometric return is created by simply adding 1 to the percentage returns and taking the product of this series over the last 5 days (the cumulative return). The geometric return for this oscillator is derived by taking subracting 1 from  the cumulative return . Then we subtract the arithmetic return from the geometric return to find the divergence. This divergence is smoothed twice using a 3-day average to create smooth signals*(note that the raw divergence produces signals in the same direction). Finally this smoothed divergence is “bounded,” by taking the PERCENTRANK of the series going back 1-year.

to be continued tommorow ……………………….

September 25, 2009 2:49 pm

Lets see if I follow the basic idea. What you are saying is that divergences between the arithmetic average and the geometric average contain directional information, right ?
If so, since the arithmetic average is always higher than the geometric, then this would be two different levels of divergence, both of the same sign, right ?
Thanks
eb

September 25, 2009 4:26 pm

hi eber, the arithmetic average is not always greater, in fact they fluctuate considerably in the short term between which is greater: AM or GM.

cheers
dv

September 26, 2009 10:29 am

dv: I am not a mathematician, and perhaps we are talking about two different things, but my understanding is that the Arithmentic Mean (AM) is always equal or greater than the Geometric Mean (GM). At least according to Wikipedia: http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means .
That article makes the point that that is true for all numbers real (this case) and positive. Not sure if we are dealing with negative numbers here. If we define daily return as Close(0) / Close (1), then they are always positive. On the other hand, if the definition is (Close(0) / Close(1)) – 1, then the returns can be negative. Which one you have in mind ?
Thanks
eb